Let Rf = ℤ[A±1] be the algebra of Laurent polynomials in the variable A and let Ra = ℤ[A±1,z 1,z2,…] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z1,z2,…. For n ≥ 1 we denote by VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(Rf)} n=1∞ and {ATLn(Ra)} n=1∞ in terms of diagrams. For each n ≥ 1 we determine presentations for both, VTLn(Rf) and ATLn(Ra). We determine sequences of homomorphisms {ρnf:Rf[VB n] →VTLn(Rf)} n=1∞ and {ρna:Ra[VB n] →ATLn(Ra)} n=1∞, we determine Markov traces {Tn′f:VTL n(Rf) → Rf} n=1∞ and {Tn′a:ATL n(Ra) → Ra} n=1∞, and we show that the invariants for virtual links obtained from these Markov traces are the f-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each n ≥ 1, the standard Temperley–Lieb algebra TLn embeds into both, VTLn(Rf) and ATLn(Ra), and that the restrictions to {TLn}n=1∞ of the two Markov traces coincide.

中文翻译：

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虚拟和箭头 Temperley–Lieb 代数、马尔可夫迹和虚拟链接不变量

让RF = ℤ[一种±1]是变量中的 Laurent 多项式的代数一种然后让R一种 = ℤ[一种±1,z 1,z2,…]是变量中的 Laurent 多项式的代数一种和变量中的标准多项式z1,z2,….为了n ≥ 1我们表示VBn虚拟编织组n股。我们定义了两个代数塔{VTLn(RF)} n=1∞和{ATLn(R一种)} n=1∞在图表方面。对于每个n ≥ 1我们确定两者的演示文稿，VTLn(RF)和ATLn(R一种). 我们确定同态序列{ρnF：RF[VB n] →VTLn(RF)} n=1∞和{ρn一种：R一种[VB n] →ATLn(R一种)} n=1∞，我们确定马尔可夫迹{吨n'F：VTL n(RF) → RF} n=1∞和{吨n'一种：ATL n(R一种) → R一种} n=1∞，我们证明了从这些马尔可夫迹中获得的虚拟链接的不变量是F-多项式用于第一道，箭头多项式用于第二道。我们证明，对于每个n ≥ 1,标准 Temperley-Lieb 代数TLn嵌入两者，VTLn(RF)和ATLn(R一种)，并且限制{TLn}n=1∞两条马尔可夫轨迹重合。